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Predictive modeling with social networks

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Predictive modeling with social networks

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Jennifer Neville & Foster Provost

Tutorial at ACM SIGKDD 2008

Tutorial

AAAI 2008

- Call networks
- Email networks
- Movie networks
- Coauthor networks
- Affiliation networks
- Friendship networks
- Organizational networks

http://images.businessweek.com/ss/06/09/ceo_socnet/source/1.htm

“…eMarketer projects that worldwide online social network ad spending will grow from $1.2 billion in 2007 to $2.2 billion in 2008, 82%.”

our focus today

- Descriptive modeling
- Social network analysis
- Group/community detection

- Predictive modeling
- Link prediction
- Attribute prediction

- Our goal is not to give a comprehensive overview of relational learning algorithms (but we provide a long list of references and resources)
- Our goal is to present
- the main ideas that differentiate predictive inference and learning in (social) network data,
- example techniques that embody these ideas,
- results, from real applications if possible, and
- references and resources where you can learn more

In three hours (less the break) we cannot hope to be comprehensive in our coverage of theory, techniques, or applications. We will present the most important concepts, illustrate with example techniques and applications, and provide a long list of additional resources.

The problem: Attribute Prediction in Networked Data

- To start, we’ll focus on the following inference problem:
- For any node i, categorical variable yi, and value c, estimate p(yi = c|DK)

DK is everything known

about the network

?

Macskassy & Provost (JMLR 2007)

provide a broad treatment

for univariate networks

Let’s start with a real-world example

- Define “Network Targeting” (NT)
- cross between viral marketing and traditional targeted marketing
- from simple to sophisticated…
- construct variable(s) to represent whether the immediate network neighborhood contains existing customers
- add social-network variables to targeting models, etc. (we’ll revisit)

- then:
- target individuals who are predicted (using the social network) to be the best prospects
- simplest: target “network neighbors” of existing customers
- this could expand “virally” through the network without any word-of-mouth advocacy, or could take advantage of it.

- Example application:
- Product: new communications service
- Firm with long experience with targeted marketing
- Sophisticated segmentation models based on data, experience, and intuition
- e.g., demographic, geographic, loyalty data
- e.g., intuition regarding the types of customers known or thought to have affinity for this type of service

Hill, Provost, and Volinsky. “Network-based Marketing: Identifying likely adopters via consumer networks. ” Statistical Science 21 (2) 256–276, 2006.

Relative Sales Rates for Marketing Segments

(1.35%)

(0.83%)

(0.28%)

(0.11%)

- Fraud detection
- Targeted marketing
- Bibliometrics
- Firm/industry classification
- Web-page classification
- Epidemiology
- Movie industry predictions
- Personalization
- Patent analysis
- Law enforcement
- Counterterrorism
- …

- The basics
- contemporary examples of social network inference in action
- what’s different about network data?
- basic analysis framework
- (simple) predictive inference with univariate networks
- disjoint inference
- network linkage can provide substantial power for inference, if techniques can take advantage of relational autocorrelation

- inductive inference (learning) in network data
- disjoint learning – models learn correlation among attributes of labeled neighbors in the network

Note on terminology: In this tutorial, we use the term “inference” to refer to the making of predictions for variables’ unknown values, typically using a model of some sort. We use “learning” to denote the building of the model from data (inductive inference). Generally we use the terminology common in statistical machine learning.Note on acronyms: see reference guide at end of tutorial

- Moving beyond the basics
- collective inference
- network structure alone can provide substantial power for inference, if techniques can propagate relational autocorrelation
- inferred covariates can influence each other

- collective learning
- learning using both the labeled and unlabeled parts of the network, requires collective inference

- social/data network vs. network of statistical dependencies
- throughout:
- example learning techniques
- example inference techniques
- example applications

- collective inference
- Additional topics (time permitting)
- methodology, evaluation, potential pathologies, understanding sources of error, other issues

So, what’s different about networked data?

- Single data graph
- Partially labeled
- Widely varying link structure
- Often heterogeneous object and link types
- From learning perspective: graph contains both training data and application/testing data

- Attribute dependencies
- Homophily; autocorrelation among variables of linked entities
- Correlation among attributes of linked entities
- Correlations between attribute values and link structure

Suggest key techniques:

guilt-by-association

relational learning

collective inference

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- Correlation between the values of the same variable on related objects
- Related instance pairs:
- Dependence between pairs of values of X:

High autocorrelation

Low autocorrelation

How can we incorporate autocorrelation into predictive inference?

Use links to labeled nodes

(i.e., guilt by association)

Thanks to (McPherson, et al., 2001)

- Birds of a feather, flock together
- – attributed to Robert Burton (1577-1640)

- (People) love those who are like themselves
- -- Aristotle, Rhetoric and Nichomachean Ethics

- Similarity begets friendship
- -- Plato, Phaedrus

- Hanging out with a bad crowd will get you into trouble
- -- Foster’s Mom

- Homophily
- fundamental concept underlying social theories
- (e.g., Blau 1977)

- one of the first features noticed by analysts of social network structure
- antecedents to SNA research from 1920’s (Freeman 1996)

- fundamental basis for links of many types in social networks (McPherson, et al., Annu. Rev. Soc. 2001)
- Patterns of homophily:
- remarkably robust across widely varying types of relations
- tend to get stronger as more relationships exist

- Now being considered in mathematical analysis of networks (“assortativity”, e.g., Newman (2003))

- fundamental concept underlying social theories
- Does it apply to non-social networks?

Biology

Functions of proteins located in together in cells (Neville & Jensen ‘02)

Tuberculosis infection among people in close contact (Getoor et al ‘01)

Business

Industry categorization of corporations that share common boards members (Neville & Jensen ‘00)

Industry categorization of corporations that co-occur in news stories (Bernstein et al ‘03)

Citation analysis

Topics of coreferent scientific papers (Taskar et al ‘01, Neville & Jensen ‘03)

Marketing

Product/service adoption among communicating customers (Domingos & Richardson ‘01, Hill et al ‘06)

Fraud detection

Fraud status of cellular customers who call common numbers (Fawcett & Provost ‘97, Cortes et al ‘01)

Fraud status of brokers who work at the same branch (Neville & Jensen ‘05)

Movies

Box-office receipts of movies made by the same studio (Jensen & Neville ‘02)

Web

Topics of hyperlinked web pages (Chakrabarti et al ‘98, Taskar et al ‘02)

- 35 K News stories

Relative Sales Rates for Marketing Segments

(1.35%)

(0.83%)

(0.28%)

(0.11%)

(Hill et al. ‘06)

What if we add in learning?

Non-relational classif.

- Logistic regression
- Neural networks
- Naïve Bayes
- Classification trees
- SVMs
- …

yi

yj

xi

xj

home location, main calling location, min of use, …

NYC,NYC,4350,3,5,yes,no,1,0,0,1,0,2,3,0,1,1,0,0,0,..

NYC,BOS,1320,2,no,no,1,0,0,0,0,1,5,1,7,6,7,0,0,1,…

BOS,BOS,6543,5,no,no,0,1,1,1,0,0,0,0,0,0,4,3,0,4,..

...

…

…

- Methods:

yi

yj

Relations

xi

xj

- Methods:

Non-relational classif.

Network classification

- Structural logistic regression
- Relational naïve Bayes
- Relational probability trees
- Relational SVMs
- …

home location, main calling location, min of use, …

NYC,NYC,4350,3,5,yes,no,1,0,0,1,0,2,3,0,1,1,0,0,0,..

NYC,BOS,1320,2,no,no,1,0,0,0,0,1,5,1,7,6,7,0,0,1,…

BOS,BOS,6543,5,no,no,0,1,1,1,0,0,0,0,0,0,4,3,0,4,..

...

…

…

- Learning where data cannot be represented as a single relation/table of independently distributed entities, without losing important information
- Data may be represented as:
- a multi-table relational database, or
- a heterogeneous, attributed graph, or
- a first-order logic knowledge base

- There is a huge literature on relational learning and it would be impossible to do justice to it in the short amount of time we have
- For additional information, see:
- Pointers/bibliography on tutorial page
- International Conference on Inductive Logic Programming
- Cussens & Kersting’s ICML’04 tutorial: Probabilistic Logic Learning
- Getoor’s ICML’06/ECML’07 tutorials: Statistical Relational Learning
- Domingos’s KDD’07/ICML’07 tutorials: Statistical Modeling of Relational Data
- Literature review in Macskassy & Provost JMLR’07

Create (aggregate) features of labeled neighbors

(Perlich & Provost KDD’03) treat aggregation

and relational learning feature construction

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Create aggregate features of relational information

Learn (adapted) flat model

Consider local relational neighborhood around instances

- where xG is a (vector of) network-based feature(s)
- Example applications:
- Fraud detection
- construct variables representing connection to known fraudulent accounts (Fawcett & Provost ‘97)
- or the similarity of immediate network to known fraudulent accounts (Cortes et al. ‘01; Hill et al. ‘06b)

- Marketing(Hill et al. ’06a)
- Creation of SN features can be (more or less) systematic: (Popescul & Ungar ’03; Perlich & Provost ’03,’06; Karamon et al. ’07,’08; Gallagher & Eliassi-Rad ‘08)
- Compare with developing kernels for relational data (e.g., Gartner’03)
- Also: Ideas from hypertext classification extend to SN modeling:
- hypertext classification has text + graph structure
- construct variables representing (aggregations of) the classes of linked pages/documents (Chakrabarti et al. ‘98; Lu & Getoor ‘03)
- formulate as regularization/kernel combination (e.g., Zhang et al. KDD’06)
- General survey of web page classification: (Qi & Davison, 2008)

- Features
- Based on boolean first-order logic features used in inductive logic programming
- Top-down search of refinement graph
- Includes additional database aggregators that result in scalar values (e.g. count, max)

- Model
- Logistic regression
- Two-phase feature selection process with AIC/BIC

- Features
- Uses set of aggregators to construct features (e.g., Size, Average, Count, Proportion)
- Exhaustive search within a user-defined space (e.g., <3 links away)

- Model
- Decision trees with probability estimates at leaves
- Pre-pruning based on chi-square feature scores
- Randomization tests for accurate feature selection (more on this later)

Recall the network marketing example…

- Features can be constructed that represent “guilt” of a node’s neighbors:
- where xG is a (vector of) network-based feature(s)
- Example application:
- Marketing(Hill et al. ’06a)

traditional variables

traditional + network

Thanks to KXEN

see also (Dasgupta et al. EDBT’08) & (Birke ’08) on social networks & telecom churn

Similar results for predicting customer attrition

Thanks to KXEN

- Use node identifiers in features connections to specific individuals can be telling

challenge: aggregation over 1-to-n relationships

- Traditional (naïve) Bayesian Classification:
- P(c|X)=P(X|c)*P(c)/P(X) Bayes’ Rule
- P(X|c)= P(xi|c) Assuming conditional independence
- P(xi|c) & P(c) Estimated from the training data

- Linked Data:
xi might be an object identifier (e.g. SSN) => P(xi|c) cannot be estimated

Let WI be a set of k objects linked to xi => P(xi|c) ~ P(linked-to-Wi|c)

P(Wi|c) ~ P(O|c) Assume O is drawn independently

P(Wi|c) ~ ( P(oj |c)) Assuming conditional independence

- Estimate from known data:
- class-conditional distributions of related identifiers (say D+ & D-)
- can be done, for example, assuming class-conditional independence in analogy to Naïve Bayes
- save these as “meta-data” for use with particular cases

- Any particular case C has its own “distribution” of related identifiers (say Dc)
- Create features
- d(Dc,D+ ), d(Dc, D- ), (d(Dc, D+ ) – d(Dc, D-))
- where d is a distance metric between distributions

- Add these features to target-node description(s) for learning/estimation
- Main idea:
- “Is the distribution of nodes to which this case is linked
- similar to that of a <whatever>?”

1: Class-conditional distributions

2: Case linkage distributions:

?

4: Extended feature vector:

3: L2 distances for C1:

L2(C1, DClass 1) = 1.125

L2(C1, DClass 0) = 0.08

Dialed-digit detector (Fawcett & P., 1997)

Communities of Interest (Cortes et al. 2001)

- nodes are people
- links are communications
- red nodes are fraudsters

these two bad guys are well connected

Estimate whether or not customer will purchase

the most-popular e-book: Accuracy=0.98 (AUC=0.96)

Class-conditional distributions across identifiers of 10 other popular books

Watch for more results later

Another unique characteristic of networked data: one can perform collective inference

Use links among unlabeled nodes

A particularly simple guilt-by-association model is that a value’s probability is the average of its probabilities at the neighboring nodes

?

- Gaussian random field (Besag 1975; Zhu et al. 2003)
- “Relational neighbor” classifier - wvRN (Macskassy & P. 2003)

- Treat network as a random field
- a probability measure over a set of random variables {X1, …, Xn} that gives non-zero probability to any configuration of values for all the variables.

- Convenient for modeling network data:
- A Markov random field satisfies
- where Ni is the set of neighbors of Xi under some definition of neighbor.
- in other words, the probability of a variable taking on a value depends only on its neighbors
- probability of a configuration x of values for variables X the normalized product of the “potentials” of the states of the k maximal cliques in the network:

(Dobrushin, 1968; Besag, 1974; Geman and Geman, 1984)

- Random fields have a long history for modeling regular grid data
- in statistical physics, spatial statistics, image analysis
- see Besag (1974)

- Besag (1975) applied such methods to what we would call networked data (“non-lattice data”)
- Some notable contemporary example applications:
- web-page classification (Chakrabarti et al. 1998)
- viral marketing (Domingos & Richardson 2001, R&D 2002)
- eBay auction fraud (Pandit et al. 2007)

- relaxation labeling – repeatedly estimate class distributions on all unknowns, based on current estimates

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- relaxation labeling – repeatedly estimate class distributions on all unknowns, based on current estimates

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- relaxation labeling – repeatedly estimate class distributions on all unknowns, based on current estimates

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- MCMC, e.g., Gibbs sampling (Geman & Geman 1984)
- Iterative classification (Besag 1986; …)
- Relaxation labeling (Rosenfeld et al. 1976; …)
- Loopy belief propagation (Pearl 1988)
- Graph-cut methods (Greig et al. 1989; …)

- estimate the maximum a posteriori joint probability distribution of all free parameters

- estimate the marginal distributions of some or all free parameters simultaneously (or some related likelihood-based scoring)

- just perform a heuristic procedure to reach a consistent state.

Using the relational neighbor classifier and collective inference, we can ask:

How much “information” is in the network structure alone?

- 12 data sets from 4 domains(previously used in ML research)
- IMDB (Internet Movie Database) (e.g., Jensen & Neville, 2002)
- Cora (e.g., Taskar et al., 2001) [McCallum et al., 2000]
- WebKB [Craven et al., 1998]
- CS Depts of Texas, Wisconsin, Washington, Cornell
- multiclass & binary (student page)
- “cocitation” links

- Industry Classification [Bernstein et al., 2003]
- yahoo data, prnewswire data

- Homogeneous nodes & links
- one type, different classes/subtypes

- Univariate classification
- only information: structure of network and (some) class labels
- guilt-by-association (wvRN) with collective inference
- plus several models
- that “learn” relational patterns

Macskassy, S. and F. P. "Classification in Networked Data: A toolkit and a univariate case study."Journal of Machine Learning Research 2007.

- network-only Bayesian classifier nBC
- inspired by (Charabarti et al. 1998)
- multinomial naïve Bayes on the neighboring class labels

- network-only link-based classifier
- inspired by (Lu & Getoor 2003)
- logistic regression based on a node’s “distribution” of neighboring class labels, DN(vi) (multinomial over classes)

- relational-neighbor classifier (weighted voting)
- (Macskassy & Provost 2003, 2007)

- relational-neighbor classifier (class distribution)
- Inspired by (Perlich & Provost 2003)

- Labeling 90% of nodes
- Classifying remaining 10%
- Averaging over 10 runs

(compare: Hill & P. “The Myth of the Double-Blind Review”, 2003)

A counter-terrorism application…

Poor concentration for primary-data only (iteration 0)

High concentration after one secondary-access phase (iteration 1)

rightmost people are

completely

unknown, therefore

ranking is uniform

(Macskassy & P., Intl. Conf. on Intel. Analysis 2005)

5046 is moderately noisy:

¼ of “known” bad guys were mislabeled

most suspicious

- high concentration of bad guys at “top” of suspicion ranking
- gets better with increased secondary-data access

- Single data graph
- Partially labeled
- Widely varying link structure
- Often heterogeneous object and link types

- Attribute dependencies
- Homophily, autocorrelation among class labels
- Correlation among attributes of related entities
- Correlations between attribute values and link structure

- Networked data can be much more complex than just sets of (labeled) vertices and edges.
- Vertices and edges can be heterogeneous
- Vertices and edges can have various attribute information associated with them

- Various methods for learning models that take advantage of both autocorrelation and attribute dependencies
- Probabilistic relational models (RBNs, RMNs, AMNs, RDNs, …)
- Probabilistic logic models (BLPs, MLNs, …)

Assume training data are fully labeled, model dependencies among linked entities

- Let’s consider briefly three approaches
- Model with inductive logic programming (ILP)
- Model with probabilistic relational model (graphical model+RDB)
- Model with probabilistic logic model (ILP+probabilities)

- The field of Inductive Logic Programming has extensively studied modeling data in first-order logic
- Although it has been changing, traditionally ILP did not focus on representing uncertainty
- First-order logic for statistical modeling of network data?
- a strength is its ability to represent and facilitate the search for complex and deep patterns in the network
- a weakness is its relative lack of support for aggregations across nodes (beyond existence)
- more on this in a minute…

- in the usual use of first-order logic, each ground atom either is true or is not true (cf., a Herbrand interpretation)

- …one of the reasons for the modern rubric “statistical relational learning”

Bill

coworkers

Amit

married

Candice

friends

- broker(Amit), broker(Bill), broker(Candice), …
- works_for(Amit, Bigbank), works_for(Bill, E_broker), works_for(Candice, Bigbank), …
- married(Candice, Bill)
- smokes(Amit), smokes(Candice), …
- works_for(X,F) & works_for(Y,F) -> coworkers(X,Y)
- smokes(X) & smokes(Y) & coworkers(X,Y) -> friends(X,Y)
- …

What’s the problem with using FOL for our task?

- Probabilistic graphical models (PGMs) are convenient methods for representing probability distributions across a set of variables.
- Bayesian networks (BNs), Markov networks (MNs), Dependency networks (DNs)
- See Pearl (1988), Heckerman et al. (2000)

- Typically BNs, MNs, DNs are used to represent a set of random variables describing independent instances.
- For example, the probabilistic dependencies among the descriptive features of a consumer—the same for different consumers

Technical

sophistication

Positive reaction

before trying service

lead user

characteristics

income

Quality

sensitivity

Positive reaction

after trying service

Amount

of use

- The term “relational” recently has been used to distinguish the use of probabilistic graphical models to represent variables across a set of dependent, multivariate instances.
- These methods model the full joint distribution over the attribute values in a network using a probabilistic graphical model (e.g., BN, MN)
- For example, the dependencies between the descriptive features of friends in a social network
- We saw a “relational” Markov network earlier when we discussed Markov random fields for univariate network data
- although the usage is not consistent, “Markov random field” often is used for a MN over multiple instances of the “same” variable

- In these probabilistic relational models, there are dependencies within instances and dependencies among instances
- Key ideas:
- Learn from a single network by tying parameters across instances of same type
- Use aggregations to deal with heterogeneous network structure

- Relational Bayesian networks
- Extend Bayes nets to network settings (Friedman et al. ‘99, Getoor et al. ‘01)
- Efficient closed form parameter estimation, but acyclicity constraint limits representation of autocorrelation dependencies and makes application of guilt-by-association techniques difficult

- Relational Markov networks
- Extension of Markov networks (Taskar et al ‘02)
- No acyclicity constraint but feature selection is computationally intensive because parameter estimation requires approximate inference
- Associative Markov networks are a restricted version designed for guilt-by-association settings, for which there are efficient inference algorithms (Taskar et al. ‘04)

- Relational dependency networks
- Extension of dependency networks (Neville & Jensen ‘04)
- No acyclicity constraint, efficient feature selection, but model is an approximation of the full joint and accuracy depends on size of training set

Example:Can we estimate the likelihood that a stock broker is/will be engaged in activity that violates securities regulations?

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- NASD (now FINRA) is the largest private-sector securities regulator
- NASD’s mission includes preventing and discovering misconduct among brokers (e.g., fraud)
- Current approach: Hand-crafted rules that target brokers with a history of misconduct (HRB)
- Task: Use relational learning techniques to automatically identify brokers likely to engage in misconduct based on network patterns

Broker

Disclosure

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Bad* Broker

Branch

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*”Bad” = having violated securities regulations

- “One broker I was highly confident in ranking as 5…
- Not only did I have the pleasure of meeting him at a shady warehouse location, I also negotiated his bar from the industry...
- This person actually used investors' funds to pay for personal expenses including his trip to attend a NASD compliance conference!
- …If the model predicted this person, it would be right on target.”
- Informal examiner feedback

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>3

CoWorkerCount(IsFraud)>3

CoWorkerCount(IsFraud)>3

DisclosureCount(Yr<1995)>3

DisclosureCount(Yr<1995)>3

DisclosureCount(Yr<1995)>3

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Broker3

Broker2

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Has Business1

Has Business3

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Is Fraud3

Is Fraud1

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DisclosureCount(Yr<2000)>0

DisclosureCount(Yr<2000)>0

DisclosureCount(Yr<2000)>0

CoWorkerCount(IsFraud)>0

CoWorkerCount(IsFraud)>0

CoWorkerCount(IsFraud)>0

DisclosureAvg(Yr)>1997

DisclosureAvg(Yr)>1997

DisclosureAvg(Yr)>1997

On Watch1

On Watch2

On Watch3

Branch1

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Learn statistical dependencies among variables

Construct “local” dependency network

Broker

Disclosure

Branch

Has Business

Region

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Is Fraud

Unroll over particular

data network for

(collective) inference

On Watch

Area

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Broker

Disclosure

Branch

Has Business

Region

Year

Is Fraud

On Watch

Area

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Broker

Disclosure

Branch

Has Business

Region

Year

Is Fraud

On Watch

Area

Type

note: needs to be “unrolled” across network

- The network of statistical dependencies does not necessarily correspond to the data network
- Example on next three slides…

Broker

Disclosure

Branch

Has Business

Region

Year

Is Fraud

On Watch

Area

Type

note: this dependency network needs to be “unrolled” across the data network

Disclosure

Statistical dependencies between brokers “jump across” branches; similarly for disclosures

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*”Bad” = having violated

securities regulations

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Broker1

Broker2

Broker3

Has Business1

Has Business3

Has Business2

Is Fraud1

Is Fraud2

Is Fraud3

On Watch1

On Watch2

On Watch3

Branch1

Region1

Area1

Disclosure1

Disclosure2

Disclosure3

Year3

Year1

Year2

Type3

Type1

Type2

(three brokers, one branch)

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>1

CoWorkerCount(IsFraud)>3

CoWorkerCount(IsFraud)>3

CoWorkerCount(IsFraud)>3

DisclosureCount(Yr<1995)>3

DisclosureCount(Yr<1995)>3

DisclosureCount(Yr<1995)>3

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Broker3

Broker2

Broker1

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Has Business1

Has Business3

Has Business2

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Is Fraud2

Is Fraud3

Is Fraud1

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DisclosureCount(Yr<2000)>0

DisclosureCount(Yr<2000)>0

DisclosureCount(Yr<2000)>0

CoWorkerCount(IsFraud)>0

CoWorkerCount(IsFraud)>0

CoWorkerCount(IsFraud)>0

DisclosureAvg(Yr)>1997

DisclosureAvg(Yr)>1997

DisclosureAvg(Yr)>1997

On Watch1

On Watch2

On Watch3

Branch1

+

–

–

Region1

+

–

+

Area1

+

DisclosureMax(Yr)>1996

DisclosureMax(Yr)>1996

DisclosureMax(Yr)>1996

+

Disclosure2

Disclosure3

Disclosure1

–

–

–

–

–

Year1

Year3

Year2

Type1

Type3

Type2

Learn statistical dependencies among variables

Construct “local” dependency network

Broker

Disclosure

Branch

Has Business

Region

Year

Is Fraud

Unroll over particular

data network for

(collective) inference

On Watch

Area

Type

- Recently there have been efforts to combine FOL and probabilistic graphical models
- e.g., Bayesian logic programs (Kersting and de Raedt ‘01), Markov logic networks (Richardson & Domingos MLJ’06)
- and see discussion & citations in (Richardson & Domingos ‘06)

- For example: Markov logic networks
- A template for constructing Markov networks
- Therefore, a model of the joint distribution over a set of variables

- A first-order knowledge base with a weight for each formula

- A template for constructing Markov networks
- Advantages:
- Markov network gives sound probabilistic foundation
- First-order logic allows compact representation of large networks and a wide variety of domain background knowledge

- A Markov Logic Network (MLN) is a set of pairs (F, w):
- F is a formula in FOL
- w is a real number

- Together with a finite set of constants, it defines a Markov network with:
- One node for each grounding of each predicate in the MLN
- One feature for each grounding of each formula F in the MLN, with its corresponding weight w

See Domingos’ KDD’07 tutorialStatistical Modeling of Relational Data for more details

Friends(A,B)

1.1

1.1

1.1

Friends(A,A)

Smokes(A)

Smokes(B)

Friends(B,B)

1.1

1.5

1.5

Cancer(A)

Cancer(B)

Friends(B,A)

Two constants:

Anna (A) and Bob (B)

wi: weight of formula i

ni(x): # true groundings of formula i in x

- Why learn a joint model of class labels?
- Could use correlation between class labels and observed attributes on related instances instead
- But modeling correlation among unobserved class labels is a low-variance way of reducing model bias
- Collective inference achieves a large decrease in bias at the cost of a minimal increase in variance

Learning helps when autocorrelation is low and there are other attributes dependencies

Learning helps when linkage is low and labeling is plentiful

- MLN/RDN/RMN:
- exploit global autocorrelation
- learning implicitly assumes training and test set are disjoint
- assumes autocorrelation is stationary throughout graph

- ACORA with identifiers (Perlich & Provost MLJ’06)
- exploits local autocorrelation
- relies on overlap between training and test sets
- need sufficient data locally to estimate

- What about a combination of the two?
- (open question)

Cora: topics in coauthor graph

IMDb: receipts in codirector graph

- Recall our network-based marketing example?
- collective inference can help for the nodes that are not neighbors of existing customers
- identify areas of the social network that are “dense” with customers

Predictive Performance

(Area under ROC curve/

Mann-Whitney Wilcoxon stat)

Predictive Performance

(Area under ROC curve/

Mann-Whitney Wilcoxon stat)

* with network sampling and pruning

Consider links among unlabeled entities during learning

- To date network modeling techniques have focused on
- exploiting links among unlabeled entities (i.e., collective inference)
- exploiting links between unlabeled and labeled for inference (e.g., identifiers)

- Can we take into account links between unlabeled and labeled during learning?
- Large body of related work on semi-supervised and transductive learning but this has dealt primarily with i.i.d. data
- Exceptions:
- PRMs w/EM (Taskar et al. ‘01)
- Relational Gaussian Processes (Chu et al. ‘06)

- Open: there has been no systematic comparison of models using different representations for learning and inference

RBN+EM

- General approach to learning arbitrary autocorrelation dependencies in within-network domains
- Combines RDN pseudolikelihood approach with mean-field approximate inference to learn a joint model of labeled and unlabeled instances
- Algorithm
- 1. Learn an initial disjoint local classiﬁer (with pseudolikelihood estimation) using only labeled instances
- 2. For each EM iteration:
- E-step: apply current local classiﬁer to unlabeled data with collective inference, use current expected values for neighboring labels; obtain new probability estimates for unlabeled instances;
- M-step: re-train local classiﬁer with updated label probabilities on unlabeled instances.

Collective learning improves performance when:(1) labeling is moderate, or (2) when labels are clustered in the network

Learning helps when autocorrelation is low and there are other attributes dependencies

Learning helps when linkage is low and labeling is plentiful

- Use unlabeled data for learning, but not for inference
- Open: No current methods do this
- However, disjoint inference is much more efficient
- May want to use unlabeled data to learn disjoint models (e.g., infer more labels to improve use of identifiers)

- Statistical tests assume i.i.d data…
- Networks have a combination of widely varying linkage and autocorrelation
- …which can complicate application of conventional statistical tests
- Naïve hypothesis testing can bias feature selection (Jensen & Neville ICML’02, Jensen et al. ICML’03)
- Naïve sampling methods can bias evaluation (Jensen & Neville ILP’03)

- Relational classifiers can be biased toward features on some classes of objects (e.g., movie studios)
- How?
- Autocorrelation and linkage reduceeffective sample size
- Lower effective sample size increases variance of estimated feature scores
- Higher variance increases likelihood that features will be picked by chance alone
- Can also affect ordering among features deemed significant because impact varies among features (based on linkage)

- Randomization tests result in significantly smaller models (Neville et al KDD’03)
- Attribute values are randomized prior to feature score calculation
- Empirical sampling distribution approximates the distribution expected under the null hypothesis, given the linkage and autocorrelation

- Within-network classification naturally implies dependent training and test sets
- How to evaluate models?
- Macskassy & Provost (JMLR’07) randomly choose labeled sets of varying proportions (e.g., 10%. 20%) and then test on remaining unlabeled nodes
- Xiang & Neville (KDD-SNA’08) choose nodes to label in various ways (e.g., random, degree, subgraph)
- See (Gallagher & Eliassi-Rad 2007) for further discussion

- How to accurately assess performance variance? (Open question)
- Repeat multiple times to simulate independent trials, but…
- Repeated training and test sets are dependent, which means that variance estimates could be biased (Dietterich ‘98)

- Graph structure is constant, which means performance estimates may not apply to different networks

- Repeat multiple times to simulate independent trials, but…

- Collective inference is a new source of model error
- Potential sources of error:
- Approximate inference techniques
- Availability of test set information
- Location of test set information

- Need a framework to analyze model systems
- Bias/variance analysis for collective inference models
- Can differentiate errors due to learning and inference processes

variance

_

Y*

bias

Y

noise

bias

variance

M1

M2

Model predictions

Test Set

M3

Samples

Models

Training

Set

learningbias

inferencebias

noise

learning bias

Y*

learning variance

_

YLI

inference variance

bias interaction term

inference bias

_

YL

Expectation over learning and inference

–

–

+

+

+

+

M1

–

–

+

+

–

–

–

+

–

–

–

+

–

–

–

M2

–

–

–

+

–

Model predictions(learning distribution)

–

–

–

+

–

–

–

–

Individual inference*on test set

+

–

+

–

–

+

M3

+

–

–

–

–

–

+

+

+

–

–

–

–

+

–

–

–

–

–

+

–

–

–

–

Samples

Learn modelsfrom samples

–

–

–

+

Training

Set

* Inference uses optimal probabilitiesfor neighboring nodes’ class labels

–

–

+

+

+

+

M1

–

–

+

+

–

–

–

+

–

–

–

+

–

–

–

M2

–

–

–

+

–

Model predictions(total distribution)

–

–

–

–

–

–

–

–

–

–

–

–

+

–

–

–

–

–

–

–

–

–

–

+

–

+

–

–

–

–

–

–

–

–

+

M3

+

–

–

–

–

–

–

–

–

–

–

–

–

–

–

+

+

+

–

–

–

–

+

–

–

–

–

–

+

–

–

–

–

Samples

Learn modelsfrom samples

–

–

–

+

Collective inferenceon test set

Training

Set

RMNs have high inference bias

RDNs have high inference variance

Loss

Bias

Variance

- Propagating label information farther in the network
- Leverage other features (Gallagher & Eliassi-Rad SNA-KDD’08)
- Create “ghost” edges (Gallagher et al. KDD’08)
- Create “similarity” edges from other features (Macskassy AAAI’07)
- Leverage graph similarity of nodes (Fouss et al. TKDE’07)
- Latent group models (Neville & Jensen ICDM’05)

- Do we know anything about the dynamics of label propagation?
- e.g., do true labels propagate faster than false ones?
- see (Galstyan & Cohen ’05a,’05b,’06,’07)

- What if labeling nodes is costly?
- Choose nodes that will improve collective inference (Rattigan et al. ‘07, Bilgic & Getoor KDD ’08)

- What if acquiring link data is costly?
- Acquire link data “actively” (Macskassy & Provost IA ‘05)

- What links you use makes a big difference
- Automatic link selection (Macskassy & Provost JMLR ‘07)
- Augment data graph w/2-hop paths (Gallagher et al. KDD ‘08)

- How does propagating information with collective inference relate to using identifiers?
- open question

- Can we identify the (causal) reason for the observed network correlation?
- Reasons might be:
- Homophily: similar nodes link together
- Social influence: linked nodes change attributes to similar values
- External factor: causes both link existence and attribute similarity

- Manski ‘93; Hill et al. ‘06; Bramoulle ‘07; Burk et al. ‘07; Ostreicher-Singer & Sundararajan ‘08; Anagnostopoulos et al. KDD’08

- Reasons might be:

- Computation and storage requirements can be prohibitive for data on real social networks -- how can we deal with massive (real) social networks?
- ignore most of the network (traditional method)
- use simple models/techniques! (e.g., Hill et al. 2007)
- reduce size of network via sampling/pruning of links and/or nodes, hopefully without reducing accuracy (much) (e.g., Cortes et al. ‘01; Singh et al. ‘05; Hill et al. ‘06b; Zheng et al. ‘07)

- What are the effects of partial network data collection?
- one may not have access to or complete control over collection of nodes and/or links
- different sampling/pruning methods may induce different effects (e.g., Stumpf et al. '05, Lee et al. '06, Handcock and Gile '02, Borgatti et al. '05)
- can we improve accuracy by sampling/pruning?
- irrelevant links/nodes can interfere with modeling (Hill et al. 2007)

- How to model networks changing over time?
- Summarize dynamic graph w/kernel smoothing (Cortes et al. ‘01, Sharan & Neville SNA-KDD’07)
- Sequential relational Markov models (Geustrin at al. IJCAI’03, Guo et al. ICML’07, Burk et al. ‘07)

- How to jointly model attributes and link structure?
- RBNs with link uncertainty (Getoor et al. JMLR‘03)
- Model underlying group structure with both links and attributes (Kubica et al. AAAI’02, McCallum et al. IJCAI’05, Neville & Jensen ICDM’05)

- Social network data often exhibit autocorrelation, which can provide considerable leverage for inference
- “Labeled” entities link to “unlabeled” entities
- Disjoint inference allows direct “guilt-by-association”
- Disjoint learning can use correlations among attributes of related entities to improve accuracy

- “Unlabeled” entities link among themselves
- Inferences about entities can affect each other (e.g., indirect gba)
- Collective inference can improve accuracy
- Results show that there is a lot of power for prediction just in the network structure
- Collective learning can improve accuracy for datasets with a moderate number of labels or when labels are clustered in the graph

- The social network can be used to create variables that can be used in traditional (“flat”) modeling
- More sophisticated learning techniques exploit networks correlation in alternative ways
- Node identifiers capture 2-hop autocorrelation patterns and linkage similarity
- Models of the joint “network” distribution identify global attribute dependencies
- These models can learn autocorrelation dependencies
- Shrinkage models can combine both local and global estimates

- Learning and evaluating network models is difficult due to complex network structure and attribute interdependencies
- There are many important methodological issues and open questions

- By this point, hopefully, you are familiar with:
- a wide-range of potential applications for network mining
- different approaches to network learning and inference
- from simple to complex

- Various issues involved with each approach
- when each approach is likely to perform well
- potential difficulties for learning accurate network models
- various methodological issues associated with analyzing network models

- Identifying groups in social networks
- Predicting links
- Entity resolution
- Finding (sub)graph patterns
- Generative graph models
- Social network analysis (SNA)
- Preserving the privacy of social networks and SNA
- Please see tutorial webpage for slides and additional pointers:http://www.cs.purdue.edu/~neville/courses/kdd08-tutorial.html

see KDD’08 tutorial: Graph Mining and Graph Kernels

see KDD’08 workshop: Social Network Mining & Analysis

Pelin Angin

Avi Bernstein

Scott Clearwater

Lisa Friedland

Brian Gallagher

Henry Goldberg

Michael Hay

Shawndra Hill

David Jensen

John Komoroske

Kelly Palmer

Matthew Rattigan

Ozgur Simsek

Sofus Macskassy

Andrew McCallum

Claudia Perlich

Ben Taskar

Chris Volinsky

Rongjing Xiang

Ron Zheng

http://pages.stern.nyu.edu/~fprovost/

http://www.cs.purdue.edu/~neville

foster provost

jennifer neville

- Birthday School year, Status (yawn?)
- Finance Conservative
- Economics Moderate
- Premed Moderate
- Politics Moderate, Liberal or Very_Liberal
- Theatre Very_Liberal
- Random_play Apathetic
- Marketing Finance
- Premed Psychology
- Politics Economics
- Finance Interested_in_Women
- Communications Interested_in_Men
- Drama Like_Harry_Potter
- Dating A_Relationship, Interested_in_Men
- Dating A_Relationship, Interested_in_Women
- Interested_in_Men&Women Very_Liberal

ACORA: Automatic construction of relational attributes (Perlich & Provost KDD'03)

AMN: Associative Markov network (Taskar ICML'04)

BN: Bayesian network

BLP: Bayesian logic program (Kersting & de Raedt '01)

DN: Dependency network (Heckerman et al. JMLR'00)

EM: Expectation maximization

GRF: Gaussian random field (Zhu et al. ICML'03)

ILP: Inductive logic programming

MLN: Markov logic network (Richardson & Domingos MLJ'06)

MN: Markov network

NT: Network targeting (Hill et al.'06)

PGM: Probabilistic graphical models

PL: Pseudolikelihood

RBC: Relational Bayes classifier (Neville et al. ICDM'03)

RBN: Relational Bayesian network (aka probabilistic relational models) (Friedman et al. IJCAI'99)

RDB: Relational database

RDN: Relational dependency network (Neville & Jensen ICDM'04)

RGP: Relational Gaussian process (Chu et al. NIPS'06)

RMN: Relational Markov network (Taskar et al. UAI'02)

RPT: Relational probability trees (Neville et al. KDD'03)

SLR: Structural logistic regression (Popescul et al. ICDM'03)

wvRN: Weighted-vector relational neighbor (Macskassy & Provost JMLR'07)